In this post (which is a follow-up to the last post), I wanted to compute the group , where is the congruence subgroup of for large enough and is prime. In fact, to make my life easier, I will also assume that and in addition, ignore -torsion. The first problem is to compute the prime to torsion. By Charney’s theorem, this will come from the cohomology of the homotopy fibre of the map

The relevant part of the Serre long exact sequence is, using classical computations of the first few K-groups of the integers together with Quillen’s computation of

Here is where it is convenient to invert primes dividing from Hurewicz theorem and Charney’s theorem we may deduce that, where denotes an equality up to a finite group of order dividing

In order to deal with -torsion, then we also have to show that the map is injective for I have a sketch of this which I will omit from this discussion but it is not too hard (assuming Quillen-Lichtenbaum). It remains to compute the homology with coefficients in . I previously computed that there was an isomorphism

where and is the adjoint representation.

Some facts concerning the cohomology of :

There are short exact sequences:

Since is annihilated by we may deduce that

as long as

Such an equality (for any group) is a claim about the Bockstein maps having a big an image as possible. Indeed, for any group there is an exact sequence:

The first and last maps here are the Bockstein maps and . Since is odd, . On the other hand, we see that the orders of the cohomology groups with coefficients in and will have the same order if and only if

Hence we have reduced to the following claim. Take the complex

where the differentials are given by the Bockstein maps. Then we have to show that the cohomology of this complex vanishes in degree two. But what are the Bockstein map is in this case? Note that since is annihilated by , the Bockstein map will be a surjective map:

To compute this explicitly, recall that the isomorphism comes from the identification of with Then, computation omitted due to laziness, we find that the Bockstein is precisely the Lie bracket. Moreover, since the (co-)homology is generated in degree one, the higher Bockstein maps can be computed from the first using the cup product formula. So the Bockstein complex above is, and I haven’t checked this because it must be true, the complex computing the mod- Lie algebra cohomology of . And this cohomology vanishes in degrees one and two, so we are done. One consequence of this computation is that

is annihilated by Moreover, the last term can be identified with the kernel of the Lie bracket (Bockstein) on

Returning to the main computation:

From the Hochschild–Serre spectral sequence and the computation of stable completed cohomology, one has an exact sequence:

From known results in characteristic zero, we immediately deduce that there is some such that there is an exact sequence

we also deduce that there is an exact sequence:

There are spectral sequences:

for and
For both of these rings, we have

Moreover, for sufficiently large we have

this follows from and is equivalent to Quillen’s computation which implies that the -groups of finite fields have order prime to Since is trivial for we deduce that

where the last equality was already used in my paper. The compatibility of the spectral sequence above for different implies that we also get an isomorphism

On the other hand, the invariant class must be an element of order in

and hence the reduction map

sends an element of order to an element of order and so

Putting things back together:

Assembling all the pieces, we see that we have proven the following:

Theorem: Let and let be sufficiently large. Let be the Lie algebra over Then, up to a finite group of order dividing we have

Moreover, still with then up to a group of order dividing we should have the same equality with replaced by